Abstract
The sunspot cycle is the magnetic cycle of the Sun produced by the dynamo process. A central idea of the solar dynamo is that the toroidal and the poloidal magnetic fields of the Sun sustain each other. We discuss the relevant observational data both for sunspots (which are manifestations of the toroidal field) and for the poloidal field of the Sun. We point out how the differential rotation of the Sun stretches out the poloidal field to produce the toroidal field primarily at the bottom of the convection zone, from where parts of this toroidal field may rise due to magnetic buoyancy to produce sunspots. In the flux transport dynamo model, the decay of tilted bipolar sunspot pairs gives rise to the poloidal field by the Babcock–Leighton mechanism. In this type of model, the meridional circulation of the Sun, which is poleward at the solar surface and equatorward at the bottom of the convection zone, plays a crucial role in the transport of magnetic fluxes. We finally point out that various stochastic fluctuations associated with the dynamo process may play a key role in producing the irregularities of the sunspot cycle.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H.D. Babcock, The Sun’s Polar Magnetic Field. Astrophys. J. 130, 364 (1959). https://doi.org/10.1086/146726
H.W. Babcock, The Topology of the Sun’s Magnetic Field and the 22-YEAR Cycle. Astrophys. J. 133, 572–587 (1961). https://doi.org/10.1086/147060
H.W. Babcock, H.D. Babcock, The Sun’s Magnetic Field, 1952–1954. Astrophys. J. 121, 349 (1955). https://doi.org/10.1086/145994
S. Basu, Global seismology of the Sun. Living Rev. Solar Phys. 13(1), 2 (2016). https://doi.org/10.1007/s41116-016-0003-4. arXiv:1606.07071 [astro-ph.SR]
A. Biswas, B.B. Karak, R. Cameron, Toroidal Flux Loss due to Flux Emergence Explains why Solar Cycles Rise Differently but Decay in a Similar Way. Phys. Rev. Lett. 129(24), 241102 (2022). https://doi.org/10.1103/PhysRevLett.129.241102. arXiv:2210.07061 [astro-ph.SR]
A. Bonanno, D. Elstner, G. Rüdiger et al., Parity properties of an advection-dominated solar alpha \(^{2}\) Omega-dynamo. Astron. Astrophys. 390, 673–680 (2002). https://doi.org/10.1051/0004-6361:20020590
B.P. Brown, M.K. Browning, A.S. Brun et al., Persistent Magnetic Wreaths in a Rapidly Rotating Sun. Astrophys. J. 711, 424–438 (2010). https://doi.org/10.1088/0004-637X/711/1/424. arXiv:1011.2831 [astro-ph.SR]
P. Caligari, F. Moreno-Insertis, M. Schüssler, Emerging flux tubes in the solar convection zone. 1: Asymmetry, tilt, and emergence latitude. Astrophys. J. 441, 886–902 (1995). https://doi.org/10.1086/175410
R.H. Cameron, J. Jiang, D. Schmitt et al., Surface Flux Transport Modeling for Solar Cycles 15–21: Effects of Cycle-Dependent Tilt Angles of Sunspot Groups. Astrophys. J. 719(1), 264–270 (2010). https://doi.org/10.1088/0004-637X/719/1/264. arXiv:1006.3061 [astro-ph.SR]
S. Chakraborty, A.R. Choudhuri, P. Chatterjee, Why Does the Sun’s Torsional Oscillation Begin before the Sunspot Cycle? Phys. Rev. Lett. 102(4), 041102 (2009). https://doi.org/10.1103/PhysRevLett.102.041102. arXiv:0907.4842 [astro-ph.SR]
S. Chandrasekhar, On the inhibition of convection by a magnetic field. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 43(340), 501–532 (1952). https://doi.org/10.1080/14786440508520205
P. Charbonneau, Dynamo Models of the Solar Cycle. Living Rev Solar Phys (2010) https://doi.org/10.12942/lrsp-2010-3
P. Charbonneau, Solar Dynamo Theory. Annu. Rev. Astron. Astrophys. 52, 251–290 (2014). https://doi.org/10.1146/annurev-astro-081913-040012
P. Charbonneau, C. St-Jean, P. Zacharias, Fluctuations in Babcock-Leighton Dynamos. I. Period Doubling and Transition to Chaos. Astrophys. J. 619, 613–622 (2005). https://doi.org/10.1086/426385
P. Charbonneau, G. Beaubien, C. St-Jean, Fluctuations in Babcock-Leighton Dynamos II. Revisiting the Gnevyshev-Ohl Rule. Astrophys. J. 658(1), 657–662 (2007). https://doi.org/10.1086/511177
P. Chatterjee, D. Nandy, A.R. Choudhuri, Full-sphere simulations of a circulation-dominated solar dynamo: Exploring the parity issue. Astron. Astrophys. 427, 1019–1030 (2004). https://doi.org/10.1051/0004-6361:20041199
F. Chen, M. Rempel, Y. Fan, Emergence of Magnetic Flux Generated in a Solar Convective Dynamo. I. The Formation of Sunspots and Active Regions, and The Origin of Their Asymmetries. Astrophys. J. 846(2), 149 (2017)
A.R. Choudhuri, The evolution of loop structures in flux rings within the solar convection zone. Solar Phys. 123, 217–239 (1989). https://doi.org/10.1007/BF00149104
A.R. Choudhuri, A correction to Spruit’s equation for the dynamics of thin flux tubes. Astron. Astrophys. 239(1–2), 335–339 (1990)
A.R. Choudhuri, Stochastic fluctuations of the solar dynamo. Astron. Astrophys. 253, 277–285 (1992)
A.R. Choudhuri, The physics of fluids and plasmas : an introduction for astrophysicists (Cambridge University Press, Cambridge, 1998)
A.R. Choudhuri, The origin of the solar magnetic cycle. Pramana 77, 77–96 (2011). https://doi.org/10.1007/s12043-011-0113-4. arXiv:1103.3385 [astro-ph.SR]
A.R. Choudhuri, The irregularities of the sunspot cycle and their theoretical modelling. Indian J. Phys. 88(9), 877–884 (2014). https://doi.org/10.1007/s12648-014-0481-y. arXiv:1312.3408 [astro-ph.SR]
A.R. Choudhuri, Nature’s third cycle: a story of sunspots (Oxford: Oxford University Press). (2015) https://doi.org/10.1093/acprof:oso/9780199674756.001.0001
A.R. Choudhuri, Starspots, stellar cycles and stellar flares: Lessons from solar dynamo models. Sci. China Phys. Mech. Astron. 60(1), 19601 (2017). https://doi.org/10.1007/s11433-016-0413-7. arXiv:1612.02544 [astro-ph.SR]
A.R. Choudhuri, A Theoretical Estimate of the Pole-Equator Temperature Difference and a Possible Origin of the Near-Surface Shear Layer. Solar Phys. 296(2), 37 (2021a). https://doi.org/10.1007/s11207-021-01784-7. arXiv:2008.02983 [astro-ph.SR]
A.R. Choudhuri, The meridional circulation of the Sun: Observations, theory and connections with the solar dynamo. Sci. China Phys. Mech. Astron. 64(3), 239601 (2021b). https://doi.org/10.1007/s11433-020-1628-1. arXiv:2008.09347 [astro-ph.SR]
A.R. Choudhuri, P.A. Gilman, The influence of the Coriolis force on flux tubes rising through the solar convection zone. Astrophys. J. 316, 788–800 (1987). https://doi.org/10.1086/165243
A.R. Choudhuri, B.B. Karak, Origin of Grand Minima in Sunspot Cycles. Phys. Rev. Lett. 109, 171103 (2012). https://doi.org/10.1103/PhysRevLett.109.171103. arXiv:1208.3947 [astro-ph.SR]
A.R. Choudhuri, M. Schüssler, M. Dikpati, The solar dynamo with meridional circulation. Astron. Astrophys. 303, L29–L32 (1995)
A.R. Choudhuri, P. Chatterjee, J. Jiang, Predicting Solar Cycle 24 With a Solar Dynamo Model. Phys. Rev. Lett. 98, 131103 (2007). https://doi.org/10.1103/PhysRevLett.98.131103. arxiv:astro-ph/0701527
M. Dikpati, P. Charbonneau, A Babcock-Leighton Flux Transport Dynamo with Solar-like Differential Rotation. Astrophys. J. 518, 508–520 (1999). https://doi.org/10.1086/307269
M. Dikpati, P.A. Gilman, Flux-Transport Dynamos with \({\alpha }\)-Effect from Global Instability of Tachocline Differential Rotation: A Solution for Magnetic Parity Selection in the Sun. Astrophys. J. 559(1), 428–442 (2001). https://doi.org/10.1086/322410
M. Dikpati, P.A. Gilman, Simulating and Predicting Solar Cycles Using a Flux-Transport Dynamo. Astrophys. J. 649, 498–514 (2006). https://doi.org/10.1086/506314
S. D’Silva, A.R. Choudhuri, A theoretical model for tilts of bipolar magnetic regions. Astron. Astrophys. 272, 621–633 (1993)
B.R. Durney, On a Babcock-Leighton dynamo model with a deep-seated generating layer for the toroidal magnetic field. Solar Phys. 160, 213–235 (1995). https://doi.org/10.1007/BF00732805
B.R. Durney, On a Babcock-Leighton Solar Dynamo Model with a Deep-seated Generating Layer for the Toroidal Magnetic Field. IV. Astrophys. J. 486, 1065–1077 (1997)
Y. Fan, Magnetic Fields in the Solar Convection Zone. Living Reviews in Solar Physics 6:4. (2009) https://doi.org/10.12942/lrsp-2009-4
Y. Fan, Magnetic fields in the solar convection zone. Living Rev. Solar Phys. 18(1), 5 (2021). https://doi.org/10.1007/s41116-021-00031-2
Y. Fan, G.H. Fisher, E.E. Deluca, The origin of morphological asymmetries in bipolar active regions. Astrophys. J. 405, 390–401 (1993). https://doi.org/10.1086/172370
M. Ghizaru, P. Charbonneau, P.K. Smolarkiewicz, Magnetic Cycles in Global Large-eddy Simulations of Solar Convection. Astrophys. J. Lett. 715(2), L133–L137 (2010). https://doi.org/10.1088/2041-8205/715/2/L133
P.A. Gilman, Dynamically consistent nonlinear dynamos driven by convection in a rotating spherical shell. II - Dynamos with cycles and strong feedbacks. Astrophys. J. Suppl. Ser. 53, 243–268 (1983). https://doi.org/10.1086/190891
L. Gizon, R.H. Cameron, M. Pourabdian et al., Meridional flow in the Sun’s convection zone is a single cell in each hemisphere. Science 368(6498), 1469–1472 (2020). https://doi.org/10.1126/science.aaz7119
G.A. Glatzmaier, P.H. Roberts, A three-dimensional self-consistent computer simulation of a geomagnetic field reversal. Nature 377(6546), 203–209 (1995). https://doi.org/10.1038/377203a0
G.A. Guerrero, J.D. Muñoz, Kinematic solar dynamo models with a deep meridional flow. Mon. Not. Roy. Astron. Soc. 350, 317–322 (2004). https://doi.org/10.1111/j.1365-2966.2004.07655.x. arxiv:astro-ph/0402097
G.E. Hale, On the Probable Existence of a Magnetic Field in Sun-Spots. Astrophys. J. 28, 315 (1908). https://doi.org/10.1086/141602
G.E. Hale, F. Ellerman, S.B. Nicholson et al., The Magnetic Polarity of Sun-Spots. Astrophys. J. 49, 153 (1919). https://doi.org/10.1086/142452
D.H. Hathaway, L. Rightmire, Variations in the Sun’s Meridional Flow over a Solar Cycle. Science 327, 1350 (2010)
G. Hazra, A.R. Choudhuri, A theoretical model of the variation of the meridional circulation with the solar cycle. Mon. Not. Roy. Astron. Soc. 472(3), 2728–2741 (2017). https://doi.org/10.1093/mnras/stx2152. arXiv:1708.05204 [astro-ph.SR]
G. Hazra, A.R. Choudhuri, A New Formula for Predicting Solar Cycles. Astrophys. J. 880(2), 113 (2019). https://doi.org/10.3847/1538-4357/ab2718. arXiv:1811.01363 [astro-ph.SR]
G. Hazra, B.B. Karak, A.R. Choudhuri, Is a Deep One-cell Meridional Circulation Essential for the Flux Transport Solar Dynamo? Astrophys. J. 782, 93 (2014). https://doi.org/10.1088/0004-637X/782/2/93
G. Hazra, A.R. Choudhuri, M.S. Miesch, A Theoretical Study of the Build-up of the Sun’s Polar Magnetic Field by using a 3D Kinematic Dynamo Model. Astrophys. J. 835, 39 (2017). https://doi.org/10.3847/1538-4357/835/1/39
H. Hotta, M. Rempel, T. Yokoyama, High-resolution Calculation of the Solar Global Convection with the Reduced Speed of Sound Technique. II. Near Surface Shear Layer with the Rotation. Astrophys. J. 798(1), 51 (2015). https://doi.org/10.1088/0004-637X/798/1/51
R. Howe, J. Christensen-Dalsgaard, F. Hill et al., Solar Convection-Zone Dynamics, 1995–2004. Astrophys. J. 634(2), 1405–1415 (2005). https://doi.org/10.1086/497107
P. Hoyng, Helicity fluctuations in mean field theory: an explanation for the variability of the solar cycle? Astron. Astrophys. 272, 321 (1993)
B.K. Jha, A.R. Choudhuri, A theoretical model of the near-surface shear layer of the Sun. Mon. Not. Roy. Astron. Soc. 506(2), 2189–2198 (2021). https://doi.org/10.1093/mnras/stab1717. arXiv:2105.14266 [astro-ph.SR]
J. Jiang, P. Chatterjee, A.R. Choudhuri, Solar activity forecast with a dynamo model. Mon. Not. Roy. Astron. Soc. 381, 1527–1542 (2007). https://doi.org/10.1111/j.1365-2966.2007.12267.x. arXiv:0707.2258
J. Jiang, R.H. Cameron, M. Schüssler, Effects of the Scatter in Sunspot Group Tilt Angles on the Large-scale Magnetic Field at the Solar Surface. Astrophys. J. 791, 5 (2014). https://doi.org/10.1088/0004-637X/791/1/5
B.B. Karak, Importance of Meridional Circulation in Flux Transport Dynamo: The Possibility of a Maunder-like Grand Minimum. Astrophys. J. 724, 1021–1029 (2010). https://doi.org/10.1088/0004-637X/724/2/1021. arXiv:1009.2479 [astro-ph.SR]
B.B. Karak, A.R. Choudhuri, The Waldmeier effect and the flux transport solar dynamo. Mon. Not. Roy. Astron. Soc. 410, 1503–1512 (2011). https://doi.org/10.1111/j.1365-2966.2010.17531.x
B.B. Karak, A.R. Choudhuri, Quenching of Meridional Circulation in Flux Transport Dynamo Models. Solar Phys. 278, 137–148 (2012). https://doi.org/10.1007/s11207-012-0142-2. arXiv:1111.1540 [astro-ph.SR]
B.B. Karak, A.R. Choudhuri, Studies of grand minima in sunspot cycles by using a flux transport solar dynamo model. Res. Astron. Astrophys. 13, 1339 (2013). https://doi.org/10.1088/1674-4527/13/11/005
B.B. Karak, J. Jiang, M.S. Miesch et al., Flux Transport Dynamos: From Kinematics to Dynamics. Space Sci. Rev. 186, 561–602 (2014). https://doi.org/10.1007/s11214-014-0099-6
B.B. Karak, L.L. Kitchatinov, A.R. Choudhuri, A Dynamo Model of Magnetic Activity in Solar-like Stars with Different Rotational Velocities. Astrophys. J. 791, 59 (2014). https://doi.org/10.1088/0004-637X/791/1/59
L.L. Kitchatinov, S.V. Olemskoy, Solar Dynamo Model with Diamagnetic Pumping and Nonlocal \(\alpha\)-Effect. Solar Phys. 276, 3–17 (2012). https://doi.org/10.1007/s11207-011-9887-2
L.L. Kitchatinov, G. Ruediger, Differential rotation in solar-type stars: revisiting the Taylor-number puzzle. Astron. Astrophys. 299, 446 (1995)
M. Küker, G. Rüdiger, M. Schultz, Circulation-dominated solar shell dynamo models with positive alpha-effect. Astron. Astrophys. 374, 301–308 (2001). https://doi.org/10.1051/0004-6361:20010686
R.B. Leighton, A Magneto-Kinematic Model of the Solar Cycle. Astrophys. J. 156, 1–26 (1969). https://doi.org/10.1086/149943
D. Longcope, A.R. Choudhuri, The Orientational Relaxation of Bipolar Active Regions. Solar Phys. 205, 63–92 (2002). https://doi.org/10.1023/A:1013896013842
D.W. Longcope, G.H. Fisher, The Effects of Convection Zone Turbulence on the Tilt Angles of Magnetic Bipoles. Astrophys. J. 458, 380 (1996). https://doi.org/10.1086/176821
L.I. Matilsky, B.W. Hindman, J. Toomre, The Role of Downflows in Establishing Solar Near-surface Shear. Astrophys. J. 871(2), 217 (2019). https://doi.org/10.3847/1538-4357/aaf647
M.S. Miesch, M. Dikpati, A Three-dimensional Babcock-Leighton Solar Dynamo Model. Astrophys. J. Lett. 785, L8 (2014). https://doi.org/10.1088/2041-8205/785/1/L8
H.K. Moffatt, Magnetic field generation in electrically conducting fluids (1978)
F. Moreno-Insertis, Nonlinear time-evolution of kink-unstable magnetic flux tubes in the convective zone of the sun. Astron. Astrophys. 166(1–2), 291–305 (1986)
A. Muñoz-Jaramillo, D. Nandy, P.C.H. Martens et al., A Double-ring Algorithm for Modeling Solar Active Regions: Unifying Kinematic Dynamo Models and Surface Flux-transport Simulations. Astrophys. J. Lett. 720, L20–L25 (2010). https://doi.org/10.1088/2041-8205/720/1/L20
D. Nandy, A.R. Choudhuri, Toward a Mean Field Formulation of the Babcock-Leighton Type Solar Dynamo. I. \(\alpha\)-Coefficient versus Durney’s Double-Ring Approach. Astrophys. J. 551, 576–585 (2001). https://doi.org/10.1086/320057
D. Nandy, A.R. Choudhuri, Explaining the Latitudinal Distribution of Sunspots with Deep Meridional Flow. Science 296, 1671–1673 (2002). https://doi.org/10.1126/science.1070955
E.N. Parker, Hydromagnetic Dynamo Models. Astrophys. J. 122, 293–314 (1955a). https://doi.org/10.1086/146087
E.N. Parker, The Formation of Sunspots from the Solar Toroidal Field. Astrophys. J. 121, 491 (1955b). https://doi.org/10.1086/146010
E.N. Parker, Cosmical magnetic fields: Their origin and their activity (Oxford University Press) (1979)
D. Passos, D. Nandy, S. Hazra et al., A solar dynamo model driven by mean-field alpha and Babcock-Leighton sources: fluctuations, grand-minima-maxima, and hemispheric asymmetry in sunspot cycles. Astron. Astrophys. 563, A18 (2014). https://doi.org/10.1051/0004-6361/201322635
S.P. Rajaguru, H.M. Antia, Meridional Circulation in the Solar Convection Zone: Time-Distance Helioseismic Inferences from Four Years of HMI/SDO Observations. Astrophys. J. 813(2), 114 (2015). https://doi.org/10.1088/0004-637X/813/2/114
M. Rempel, Flux-Transport Dynamos with Lorentz Force Feedback on Differential Rotation and Meridional Flow: Saturation Mechanism and Torsional Oscillations. Astrophys. J. 647, 662–675 (2006). https://doi.org/10.1086/505170
K. Schatten, Fair space weather for solar cycle 24. Geophys. Rev. Lett. 32(21), L21106 (2005). https://doi.org/10.1029/2005GL024363
K.H. Schatten, P.H. Scherrer, L. Svalgaard et al., Using Dynamo Theory to predict the sunspot number during Solar Cycle 21. Geophys. Rev. Lett. 5(5), 411–414 (1978). https://doi.org/10.1029/GL005i005p00411
H. Schwabe, Sonnenbeobachtungen im Jahre 1843. Von Herrn Hofrath Schwabe in Dessau. Astronomische Nachrichten 21(15), 233 (1844). https://doi.org/10.1002/asna.18440211505
H.C. Spruit, Equations for thin flux tubes in ideal MHD. Astron. Astrophys. 102, 129–133 (1981)
M. Steenbeck, F. Krause, K.H. Rädler, Berechnung der mittleren Lorentz-Feldstärke v X B für ein elektrisch leitendes Medium in turbulenter, durch Coriolis-Kräfte beeinflußter Bewegung ( A calculation of the mean electromotive force in an electrically conducting fluid in turbulent motion, under the influence of Coriolis forces). Zeitschrift Naturforschung Teil A 21, 369–376 (1966)
J.O. Stenflo, Magnetic-Field Structure of the Photospheric Network. Solar Phys. 32(1), 41–63 (1973). https://doi.org/10.1007/BF00152728
J.O. Stenflo, A.G. Kosovichev, Bipolar Magnetic Regions on the Sun: Global Analysis of the SOHO/MDI Data Set. Astrophys. J. 745, 129 (2012). https://doi.org/10.1088/0004-637X/745/2/129
L. Svalgaard, E.W. Cliver, Y. Kamide, Sunspot cycle 24: Smallest cycle in 100 years? Geophys. Rev. Lett. 32(1), L01104 (2005). https://doi.org/10.1029/2004GL021664
S. Tobias, D. Hughes, N. Weiss, Unpredictable Sun leaves researchers in the dark. Nature 442(7098), 26 (2006). https://doi.org/10.1038/442026c
S. Tsuneta, K. Ichimoto, Y. Katsukawa et al., The Magnetic Landscape of the Sun’s Polar Region. Astrophys. J. 688(2), 1374–1381 (2008). https://doi.org/10.1086/592226
I.G. Usoskin, S.K. Solanki, G.A. Kovaltsov, Grand minima and maxima of solar activity: new observational constraints. Astron. Astrophys. 471(1), 301–309 (2007). https://doi.org/10.1051/0004-6361:20077704
Y.M. Wang, A.G. Nash, N.R. Sheeley Jr., Magnetic flux transport on the sun. Science 245, 712–718 (1989). https://doi.org/10.1126/science.245.4919.712
Y.M. Wang, J.N.R. Sheeley, A.G. Nash, A New Solar Cycle Model Including Meridional Circulation. Astrophys. J. 383, 431 (1991). https://doi.org/10.1086/170800
N.O. Weiss, Convection in an imposed magnetic field. Part 1. The development of nonlinear convection. J. Fluid Mech. 108, 247–272 (1981). https://doi.org/10.1017/S0022112081002115
N.O. Weiss, F. Cattaneo, C.A. Jones, Periodic and aperiodic dynamo waves. Geophys. Astrophys. Fluid Dynam. 30(4), 305–341 (1984). https://doi.org/10.1080/03091928408219262
A.R. Yeates, A. Muñoz-Jaramillo, Kinematic active region formation in a three-dimensional solar dynamo model. Mon. Not. Roy. Astron. Soc. 436, 3366–3379 (2013)
H. Yoshimura, Solar-cycle dynamo wave propagation. Astrophys. J. 201, 740–748 (1975). https://doi.org/10.1086/153940
Acknowledgements
I thank two anonymous referees for very valuable comments on an earlier version of the paper, which helped in improving the paper. I acknowledge financial support from the Contingency Grant of the Honorary Professorship offered to me by the Indian Institute of Science.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
I hereby declare that this paper has no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Choudhuri, A.R. The emergence and growth of the flux transport dynamo model of the sunspot cycle. Rev. Mod. Plasma Phys. 7, 18 (2023). https://doi.org/10.1007/s41614-023-00120-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s41614-023-00120-9